wanminliu@gmail.com
Ext
open
boundary
•
¥
6
-
-
-
Her
oin
-
µ
lEy=
x
The
btD
is
not
in
D.
y
=
✗
&
yz
o
Y=
-
✗
&yz
o--
A-
☐
:
I
;
¥×
]
y
so
g.
ix.
YI
has
meany
on
D
.
wanminliu@gmail.com
=
¥
what's
ds
⑤
¥
,
#
ds
"
if
the
syne
is
giv
en
by
a
gr
aph
2-
=
2-
ix.
YI
,
wher
e
4=1
ix.
y
,
HEIR
}
/
then
ds
=
i-
zi-ayt
dx
dy
.x
-yi-
z
-l.XZt
a.
t
o/'#i+y'+
z'--
1
.
and
a
is
the
distant
so
we
can
whet
fr
om
the
observ
er
C-
the
cent
er
✗
=
✗
142-
1
,
4=4
7=2
of
the
moon
.
we
use
Y
and
2-
as
paramet
er
so
ds=*j+
x
dy
dz
✗
=
-'
Sime
✗
is
positiv
e
¥¥
É÷
.
✗
y
=
I
(
1-
y
'
-
2-25
?
C-
4)
=
-
y
a-
yIzj±
✗
It
✗
z
=
÷
(
1-
y
?
2-
'
-22-
1=-2-
11
-
y
?
z
Y
¥
wanminliu@gmail.com
Ti
e
n
¥
1
!
ds=
¥
H÷
÷_
pqgt
o-
YF
.net
dS=ÑÉx
dy
dz
Next
,
we
giv
e
a
description
of
=
.
dy
dt
te
p
w
j
e
t
m
g-
Y
to
tie
7-
7-
plane
Y
=
rco
so
2-
=
rsino
o
e
re
t
-
Eso
EI
=
dy
da
ñ
I
=
¥µ
ay
dx
=
¥
1
.
1.
±
.
=
¥
.
-4
"d
.
a
0
1-
F-
u
arms
fr om
tai
l
9-
→
=
f-
fa
t
.
d÷=&
WE
/
=
-441
-
E)
☐
wanminliu@gmail.com
why
dS=HZ
×
i-Ñdx
dy
?
'
¥
-
let
the
surf
er
by
Ñcx
-
y
)
=
IX.
Y
,
2-
ix.
Y
)
)
→
p
×
Then
Ñx
1
×
-77
=
11
,0
,
2-
x
)
@
=
angle
between
r}
IX.
Y
)
=
10
,
I
,
2-7
)
rI
and
try
*
xril-lrillriis.mil
e-
=
ri.ru
?
--F
xlY
riill-wi0
1
a
-
-
5.
ri
=
Kii
-
lriiryiuioe-ri.rs
=*
.ñlñ•ñ
-
lñ÷÷
=
T=
I
-
So
ds
=/
Fiery
/
=dx
dy=Ea
dx
dy
wanminliu@gmail.com
If
ÑC
X
,
7)
=
I
✗
.
Y
,
2-
IX.
Y
)
)
then
ri
=
11
.
0
,
2-
x
)
try
=
10
,
I
,
Zy
)
so
=L
=
RI
-
RI
=
(
1,
0
,
2
×
1
.
I
1.
0
.
Zx
)
=
It
2-
I
G-
=
try
.
Tr
y
=
I
0. 1 . 7
,
)
-1
1
0
,1
,
24
1
=
1
1
-
2
-
5
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
&
So
-Ñ=
µ
xx) ft
ty
J-lZxZ
=lt
ai+
zi-Ézaiy#
wanminliu@gmail.com
By
this
method
,
"
"
"
"
4-
we
pick
the
surf
ace
F-
=
rj.r
o
?
as x
--
XYY
.Z)G--Ir
.rI
go
the
svfneitShtn
b
Y
.E@I
z
-
1-
xyi-
x
a
T
r
TY
.
t_lXlY
.Z1
.
Y
,
2-
)
try
=
I
Xy
,
I
,
0
)
Pz
=
I
✗
2-
,
0
,
I
7
wanminliu@gmail.com
Pe
t
R
be
the
re
g
i
o
n
bounded
by
the
①
comput
e
1St
xy
dv
or
by
using
the
cylinder
coor
dinate
cylinder
5+5=1
and
the
planes
2-
=
1
,
7=0
S
¥
.
o
ylinder
coodtnaÉ
in
the
fo
n
t
octant
(
r
,
O
.
2-
)
coor
dinate
in
the
plane
f-
DV
=
ds
.
It
✗
=
roo
m
_#
=
"
rd
rd
c
i
d
z
-
Y
-
r
s
h
o
-
x
d
-
rd
r
-
r
f
o
o
i
.
n
o
)
÷÷i÷"
¥
✗
<
a
wanminliu@gmail.com
=
if
&
-
I
③
comput
e
fssxy
dy
or
by
Gaius
div
er
gence
Theor
em
.
Gauss
div
er
gence
theor
em
.
f-
-4
!
"
"
)
¥
gggg
,
w=
☒
É*s
=
f-
E)
fa
s
t
.
-
ciao
-
É
-
ix.
=
-1
-
4
1
×
1
-
2
1
✗
É
=
I
r
s
☐
wanminliu@gmail.com
S
=
S
,
U
Seuss
vsx
Uss
we
need
a
ve
r
t
u
field
É
is
the
boon
day
not
R
.
go
that
dw
=p
=
XY
Fo
r
eah
Si
,
we
can
denote
its
unit
normal
ve
to
by
Ñ
,
we
can
pick
up
É=
OF
-1
0
5
-1
xx
É
.
-
-
-
-
,
?
abaef
dises-divf
-C
-
x.
g-y
.b
z
)
(0
, 9
×
77)=117
.
So
-
in
#
-
plane
:*
:;
¥ ¥
± :
¥
É÷
É
¥
÷
.
am
Ifl
div
ed
V
=
☒
E.
Ñds
n
s
✗
5g
(
fr
ont
cylinder
)
S
,
-
bottom
Edina
wanminliu@gmail.com
Then
Ñ
,
=
(
0,
0
,
-1
)
Iff
divÉdv=ÉifÉ.Ñds
Ñ
,
=
(
o
,
o
,
1)
n
si
Ñ
,
=
I
-1
,
0,
0
)
the
3r
d
component
"
"
"
=
"
"
"
"
"
Ñ
NJ
=L
,
,
0
)
is
0
.
became
His
so
=P
.
Ñi
=
0
1=13
,4
,
5
in
the
✗
y
-
plane
É
.
Ñz
=
Xy
z
/
(
2-
=
const
)
.
Fo
r
5
,
,
=
10
.0.0
)
.
2-
=/
=D
xy
ds
=
If
✗
Yd
s
=)
!
f÷
r
>
saoaodr
do
=
f-
q
✗
Ey
's
/
"
✗
→
%
armoir
e
,
y=rs W
osos
E
☐
(7--
1)
wanminliu@gmail.com
¥
-4k
be
the
intersection
of
thigh
theorist
Kr
oupa
.
.
and
origin
is
the
the
plane
9=7
and
the
spher
e
center
of
the
spher
e
✗
Yy
'
-1
2
-1
-1
Sette
inter
sects
in
the
dir
ector
count
er
clockwise
is
a
big
cir
cle
.
with
re
s
p
e
c
t
to
2-
-
axis
.
1
×
45+2-2=1
1
×
7-22-41
①
comput
e
the
int
egr
al
7=2
Y=
2-
✗
=
WI
§
xyz
dz
|
✗
=
v0
g
,
nanny
.my
,
gun
,
,
÷
?
g
y
,
÷
,
,
,
•
µ
,
,
1-
2-
=
¥
since
It
=
¥
cow
do
②
comput
e
the
same
int
egral
so
§
✗
Hdz
=/
o
"
v0
ftp
.sina/Krsinojtrir
od0
1-
by
st
ock
s
's
theor
em
.
=
z-
f
.
f
i
"
@
sojlsmojdo
wanminliu@gmail.com
22
=
¥
#
f.
Eggnog
.
do
¥
met
€
We
wan t
to
use
st
ock
s
:
Theon
5h20
☒
we
pick
up
the
ve
r
tov
field
|
fg
y
n
z
o
j
=
¥
40
É
=
(
o
,
o
,
XYZ
)
d
}
=L
dx.
dy
.
dz
)
=
¥
1
.
"
;
so
so
☒
•
DE
=
Xy
z
dz
=
*
.
£
.
2k
=
a
§
.
di
=
If
☐
✗
E.
Ñds
1-
9
s
st
ock
s
's
Theor
em
wanminliu@gmail.com
So
Ñ=
10
.
¥
)
wher
e
the
boundy
of
the
Sunfir
e
S
is
the
closet
yxÉ=anlÉ
ii.
I
curv
e
T
.
i
suf
o
S
/
o
,
o
,
XYZ
s
"
*
=/
÷
:|
0*+
14-
12=0
=
2K
¥
T
-
3
×
1
×
42-
1
'T
a
normal
dtvatn
{
is
(
0
,
I
,
-
1)
=
IXZ
,
-77
,
0
)
the
plane
2=4
or
10
,
-1
,
a)
So
✗
E)
•
Ñ
we
need
the
3-
rd
u
m
p
w
t
t
be
2=1
×
7
,
-47
,0110
,
-
¥
,
¥
)
polka
,
so
we
pick
up
Ñ=
=
7
¥
4
unit
normal
ve
r
tu
fe
u
d
wanminliu@gmail.com
I
}
✗
E)
•
Ñds
=
If
as
ñ+y
:-c
7=2
S
The
plane
z=
y
.
SIT
'd
×
d
y
µ
÷
w Y
,
¥
X'
+
visa
dS=HZ
×
4 Ñ=dxdy
=
,+ d
×
q= rÉ
⇐
""
¥
"
am
§
¥
ds=ss÷
→
.
,
÷÷÷÷÷÷
=
a
¥¥
☐